Elliptical instability of a vortex tube and drift current induced by it

Turbulent Mixing and BeyondTurbulent Mixing and Beyond International ConferenceInternational Conference　　　　 August 18-26, 2007 (Aug. 24)August 18-26, 2007 (Aug. 24)

The Abdus Salam International Centre The Abdus Salam International Centre for Theoretical Physics (ICTP)for Theoretical Physics (ICTP) 　　　　 Trieste, ItalyTrieste, Italy

Yasuhide Fukumotoand

Makoto Hirota

Graduate School of Mathematics, Kyushu University, Fukuoka, Japan

Aircraft trailing vortices (Higuchi

1993)　　　　　

Cessna Citation IV from B25Cessna Citation IV from B25

Instability of trailing vortices

(Crow 1970)　　　　　

from from Van Dyke:Van Dyke:An Album of Fluid MotionAn Album of Fluid Motion

B-47B-47

Short-wave Instability of trailing vortices 　　　　

(Leweke & Williamson ’98)(Leweke & Williamson ’98)

Axial flow in a vortex ring

Naitoh, Fukuda, Gotoh,Naitoh, Fukuda, Gotoh,Yamada & Nakajima Yamada & Nakajima (’02)(’02)

cf.cf. Maxworthy Maxworthy (’77)(’77)

Contents1. Introduction

2. Influence of a pure shear on Kelvin waves “A global stability of the Rankine vortex to three-dimensional disturbances" Moore & Saffman ('75), Tsai & Widnall ('76) Eloy & Le Dizès ('01), Y. F. ('03)

　　　　　　　　 elliptical instability (local stability)

cf. vortex ring: Hattori & Y. F. ('03), Y. F. & Hattori ('05)

3. Energy of Kelvin waves Cairn’s formula (’79), Y. F. ('03), Hirota & Y. F. ('07) for continuous spectra

4. Weakly nonlinear corrections to Kelvin waves kinematically accessible variations (= isovortical perturbations)

　　 → drift current

Elliptically strained vortex

Expand infinitesimal disturbance in Suppose that the core boundary is disturbed toSuppose that the core boundary is disturbed to

the linearized Euler equationsthe linearized Euler equations

Example of a Kelvin wave m=4

Dispersion relation of Kelvin waves 1m

m=-1m=-1 (solid lines) and (solid lines) and m=1m=1 (dashed lines) (dashed lines)

Equations for disturbance of )(O

Solution of disturbance of

)(O

For the For the mm wave, we find, from the Euler equations, wave, we find, from the Euler equations,

andand

(radial wave numbers)(radial wave numbers)

Disturbance field is explicitely written out.Disturbance field is explicitely written out.

)(O

Growth rate of helical waves (m=±1)

Instability occurs at Instability occurs at everyevery intersection points intersection points of dispersion curves of of dispersion curves of ((m, m+2m, m+2) waves ) waves ???

Krein’s theory of Hamiltonian spectra

Spectra of a Spectra of a fintefinte-dimensional Hamilton system-dimensional Hamilton system

Energy of a Kelvin wave

(averaged)(averaged)Excess energyExcess energy for generating for generating

a Kelvin wavea Kelvin wave

base flowbase flow disturbancedisturbance

Kelvin waveKelvin wave stationarystationary component ??? component ???(no strain)(no strain)

Carins’ formula (Carins ‘79)

Energy of a helical wave (m=1)

Energy signature of helical waves (m=±1)

m=-1m=-1 (solid lines) and (solid lines) and m=1m=1 (dashed lines) (dashed lines)

Difficulty in Eulerian treatment

Excess energyExcess energy

base flowbase flow disturbancedisturbance

Complicated calculation would be required forComplicated calculation would be required for

Steady Euler flows

iso-vortical sheetsiso-vortical sheets

Kinematically accessible variationKinematically accessible variation (= (= preservation of circulationpreservation of circulation))

TheoremTheorem (Kelvin, Arnold ’66)(Kelvin, Arnold ’66) A steady Euler flow is a A steady Euler flow is a 　　 coditional extremum of energy coditional extremum of energy HH onon an iso-vortical sheean iso-vortical sheett 　　 (=(= w.r.t.w.r.t. kinematically accessible variations kinematically accessible variations).).

Variational principle for stationary vortical region

☆☆ Volume preservingVolume preserving displacement of fluid particles: displacement of fluid particles:

☆☆ Iso-vorticity:Iso-vorticity:

Then. usingThen. using

First and second variationsThe first variationThe first variation

Given which satisfies Given which satisfies

ThenThen is a solution ofis a solution of

The secobd variationThe secobd variation

Further, given which satisfies Further, given which satisfies

ThenThen is a solution ofis a solution of

( : projection operator )( : projection operator )

Wave energy in terms of iso-vortical disturbance

Excess energyExcess energy

by by Arnold’s theoremArnold’s theorem

It is proved thatIt is proved that

and thatand that

is the wave-energyis the wave-energy

does not contribute todoes not contribute to

are are linearlinear disturbances!!disturbances!!

Drift currentTake the average over a long timeTake the average over a long time

For the For the Rankine vortexRankine vortex

Substitute the Substitute the Kelvin wave Kelvin wave

• There is no contribution fromThere is no contribution from• For For 2D2D wave, wave, genuinly genuinly 3D3D effect effect !! !!

Drift current caused by Kelvin waves

Displacement vector of Displacement vector of mm wave wave

Flow-flux, of Flow-flux, of mm wave, in the wave, in the axial axial directiondirection

Axial flow-flux of buldge wave (m=0), elliptic wave (m=2)

m=0m=0 (dashed lines) and (dashed lines) and m=2m=2 (solid lines) (solid lines)

Dispersion Dispersion relationrelation

For the For the principal principal mode,mode,

• 1.242, -1.2421.242, -1.242• 3.370, -0.24433.370, -0.2443• 7.058, -0.090467.058, -0.09046• 8.882, -0.068288.882, -0.06828• 12.521, -0.0456412.521, -0.04564

Axial flow-flux of a helical wave (m=1)

For the For the principal principal modemode 　（＝　（＝ stationary)stationary)

• 2.5052.505• 4.3494.349

Axial current of staionary helical modes )1( m

For stationary modesFor stationary modestime average is not necessary :time average is not necessary :

Given,Given,

Summary

1.1. Tsai & Widnall ('76) is simplified; Tsai & Widnall ('76) is simplified; Disturbance field and growth rate Disturbance field and growth rate are written out in terms of the Bessel and modified Bessel functions.are written out in terms of the Bessel and modified Bessel functions.

2.2. Energetics: Energy of the Kelvin waves is calculated by adapting Energetics: Energy of the Kelvin waves is calculated by adapting CairCairns’ formulans’ formula (= (= black boxblack box)) 　 　 consistent with consistent with Krein’s theoryKrein’s theory

Linear stability of an Linear stability of an elliptic vortexelliptic vortex, a straight vortex tube subject to a , a straight vortex tube subject to a pure shear, to pure shear, to three-dimensionathree-dimensional disturbances is calculated.l disturbances is calculated.This is a parametric resonance instability between two Kelvin waves This is a parametric resonance instability between two Kelvin waves caused by a perturbation breaking S-symmetry of the circular core.caused by a perturbation breaking S-symmetry of the circular core.

Modification of mean field at 2 nd orderModification of mean field at 2 nd order ：：

3. 3. Lagrangian approachLagrangian approach: Energy of the Kelvin waves is calculated by rest: Energy of the Kelvin waves is calculated by restricting disturbance to ricting disturbance to kinematically accessible fieldkinematically accessible field

linear linear perturbation is sufficient to calcilate energy, quadratic in amplitude!perturbation is sufficient to calcilate energy, quadratic in amplitude!

4. 4. Axial currentAxial current: For the : For the Rankine vortexRankine vortex, 2 nd-order drift current incl, 2 nd-order drift current includes not only azimuthal but also udes not only azimuthal but also axialaxial component component